Posets uniquely determined by its compact saturated subsets

TL;DR

The paper studies when a poset or dcpo is uniquely determined by the lattice of its Scott compact saturated subsets, introducing the notion of $\mathcal{Q}_{\sigma}$-uniqueness. It proves that quasicontinuous domains yield $\mathcal{Q}_{\sigma}$-unique posets and identifies a class of $K_D$ posets (with co-sobriety) that are also $\mathcal{Q}_{\sigma}$-unique, expanding to dcpos via conditions involving co-sobriety and directed supersets of quasicontinuous upper cones. It further shows that weakly well-filtered co-sober posets are $\mathcal{Q}_{\sigma}$-unique and demonstrates through examples that the several sufficient conditions are distinct and not all necessary, thereby highlighting the nuanced landscape of $\mathcal{Q}_{\sigma}$-uniqueness. The work concludes with open questions about the necessity of co-sobriety and seeks a complete characterization of $\mathcal{Q}_{\sigma}$-unique posets and dcpos.

Abstract

Inspired by Zhao and Xu's study on which a dcpo can be determined by its Scott closed subsets lattice, we further investigate whether a poset (or dcpo) $P$ is able to be determined by the family $\mathcal Q(P)$ of its Scott compact saturated subsets, in the sense that the isomorphism between $(\mathcal Q(P), \supseteq)$ and $(\mathcal Q(M), \supseteq)$ implies the isomorphism between $P$ and $M$ for any poset (or dcpo) $M$, in such case, $P$ is called $\mathcal Q_σ$-unique. Quasicontinuous domains are proved to be $\mathcal Q_σ$-unique posets and draw support from which, we provide a class of $\mathcal Q_σ$-unique dcpos. We also define a new kind of posets called $K_D$ and show that every co-sober $K_D$ poset is $\mathcal Q_σ$-unique. It even yields another kind of $\mathcal Q_σ$-unique dcpos. It is gratifying that weakly well-filtered co-sober posets are also $\mathcal Q_σ$-unique. At last, we distinguish among the conditions which make a poset (or dcpo) $\mathcal Q_σ$-unique from each other by some examples; meanwhile, it is confirmed that none of them except the property of being co-sober are necessary for a poset (or dcpo) to be $\mathcal Q_σ$-unique.

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Theorems & Definitions (34)

• Definition 2.1
• Remark 2.2
• Example 2.3
• Definition 2.4
• Definition 3.1
• Definition 3.2
• Example 3.3
• Lemma 3.4
• Theorem 3.5
• proof